In view of numerous experimental results reported in the past decades on the observation of faster-than-light electromagnetic signals, we analyze the structure of relativistic kinematics, where such superluminal signals are allowed. As the first step, we suggest replacing the Einstein postulates with the general relativity principle (the possibility of describing any phenomenon in any frame of reference achievable in nature) applied to an inertial motion in an empty space. Then, as in common relativistic kinematics, we also arrive at the Lorentz transformations between inertial reference frames, where a superluminal motion of massless entities is not prohibited (in particular, for perturbations of bound electromagnetic field). However, for any objects with a finite rest mass, the limited velocity remains always less than the light velocity c, and in such a way we avoid the tachyonic-type theories in their common meaning. We show that the application of superluminal electromagnetic signals to synchronization of distant clocks yields the common expressions for the relativity of the simultaneity of events for different inertial observers. This result confirms the validity of the Lorentz transformations in generalized relativistic kinematics, though along with superluminal signals. Hence we arrive at the invariance of the space-time interval, as in common relativistic kinematics, where, however, the superluminal motion of massless entities is allowed. Even so, no further changes emerge in relativistic dynamics and other common relativistic implications. Finally, we consider causal paradoxes related to the propagation and exchange of superluminal signals between inertial observers and provide their resolution.